The purpose of this note is to improve the bound in theorem:gowersbilinear to \(r= \log ^{O(1)} \alpha ^{-1}\) , as was conjectured in [1]}. Our proof is arguably simpler and is obtained only by invoking theorem:sanders a few times, without doing any extra Fourier analysis. The motivation behind this work — other than obtaining a bound that matches the linear case — is to employ this result in a more algebraic framework to obtain a modular and simpler proof of a \(U^4\)  inverse theorem. Moreover, theorem:main was recently used by Bienvenu and Lê [2]} to obtain upper bounds on the correlation of the Möbius function over functions fields with quadratic phases.
